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Contract No. DE-AC36-08GO28308

Comparison of Second Order 

Loads on a Semisubmersible 

Floating Wind Turbine

Preprint

S. Gueydon

MARIN

T. Duarte

EDP Inovação

J. Jonkman

National Renewable Energy Laboratory

I. Bayati 

Dipartimento di Meccanica, Politecnico di Milano 

A. Sarmento 

Instituto Superior Técnico, University of Lisbon

To be presented at the 33rd International Conference on Ocean, 

Offshore and Arctic Engineering

San Francisco, California

June 8 – 13, 2014

Conference Paper

NREL/CP-5000-61162

March 2014
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COMPARISON OF SECOND-ORDER LOADS ON A SEMISUBMERSIBLE FLOATING

WIND TURBINE

Sébastien Gueydon

MARIN

Wageningen, Netherlands

Tiago Duarte

EDP Inovação

Instituto Superior Técnico

Lisbon, Portugal

Jason Jonkman

National Renewable Energy 

Laboratory 

Golden, Colorado USA

Ilmas Bayati

Dipartimento di Meccanica

Politecnico di Milano

Milano, Italy

António Sarmento

Instituto Superior Técnico

Lisbon, Portugal

ABSTRACT

As offshore wind projects move to deeper waters, floating 

platforms become the most feasible solution for supporting the 

turbines. The oil and gas industry has gained experience with 

floating platforms that can be applied to offshore wind projects.

This paper focuses on the analysis of second-order wave 

loading on semisubmersible platforms. Semisubmersibles, 

which are being chosen for different floating offshore wind 

concepts, are particularly prone to slow-drift motions. The slack 

catenary moorings usually result in large natural periods for

surge and sway motions (more than 100 s), which are in the 

range of the second-order difference-frequency excitation force.

Modeling these complex structures requires coupled design 

codes. Codes have been developed that include turbine 

aerodynamics, hydrodynamic forces on the platform, restoring 

forces from the mooring lines, flexibility of the turbine, and the 

influence of the turbine control system. In this paper two 

different codes are employed: FAST, which was developed by 

the National Renewable Energy Laboratory, and aNySIM, 

which was developed by the Maritime Research Institute 

Netherlands. The hydrodynamic loads are based on potentialflow theory, up to the second order. Hydrodynamic coefficients 

for wave excitation, radiation, and hydrostatic forces are 

obtained with two different panel codes, WAMIT (developed by 

the Massachusetts Institute of Technology) and DIFFRAC

(developed by MARIN). 

The semisubmersible platform, developed for the 

International Energy Agency Wind Task 30 Offshore Code 

Comparison Collaboration Continuation project is used as a 

reference platform. Irregular waves are used to compare the 

behavior of this platform under slow-drift excitation loads. The

results from this paper highlight the effects of these loads on 

semisubmersible-type platforms, which represent a promising 

solution for the commercial development of the offshore 

deepwater wind resource.

NOMENCLATURE

dS surface element of the wetted hull m2

dl element of the water line contour m 

F(2) second-order wave forces kN 

g gravity constant m/s2

M mass matrix of the body t, tm, tm2

n0 outward pointing normal vector - 

O origin of the underwater geometry at s.w.l.

t time s 

S wetted surface of the hull m2

s.w.l. still water line

X 6 component vector with positions of point O

X(1) surge m 

X(2) sway m 

X(3) heave m 

X(4) roll rad

X(5) pitch rad

1

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X(6) yaw rad

X second time derivative of vector X m⋅s

–2, rad⋅s–2 

ζrel relative wave elevation m 

φ velocity potential m2/s

ρ mass density of water kg/m3

Ω angular motion vector rad

ωj j-th wave frequency rad/s

W.L. Water Line

INTRODUCTION

The need for clean sources of renewable energy has driven 

the development of new technologies. Wind energy has become

a readily available solution for onshore locations, and offshore 

wind turbine installations are expanding. The chosen offshore 

sites provide a better wind resource, with less turbulence and 

higher wind speeds. Newly designed multimegawatt wind 

turbines are now undergoing prototype tests, with nominal 

power in the range of 5 to 8 MW and rotor diameters up to 164

m.

Most of the offshore wind farms are located in shallow 

water (<30 m), where fixed-bottom foundations are the most 

feasible solution. The turbines are mainly mounted on monopile 

structures, and most of the capacity is installed in the North 

Sea. Shallow water locations, however, are limited, and most of 

the offshore wind resource is located in deeper waters, where 

fixed-bottom solutions become unfeasible. For this reason, 

floating platforms are being developed to support the new 

multimegawatt wind turbines in water depths greater than 50 m. 

Significant research and development efforts, which rely on 

numerical tools and experimental studies, are ongoing. Two 

full-scale prototypes are already operating in Europe (the 

Hywind and WindFloat prototypes). 

In this context, the development of floating wind turbines 

relies on properly predicting fatigue and extreme loads on the 

turbine, its floating substructure, and its mooring system. Such 

predictions are possible only with simulations that account

simultaneously for the hydrodynamics loads and response of 

the floater, and the aeroelastic load and response of the turbine. 

The computer program FAST, developed by the National 

Renewable Energy Laboratory (NREL), is one of the tools 

offering this functionality. Recently the software aNySIM, 

developed by the Maritime Research Institute Netherlands

(MARIN), has been coupled to PHATAS (developed by Energy 

research Centre of the Netherlands) to also enable the 

simulation of a floating wind turbine [1]. The results of FAST 

and aNySIM for a semisubmersible platform are compared in 

this study with a focus on second-order wave loads. The 

semisubmersible of the Offshore Code Comparison 

Collaboration Continuation (OC4) project, which operated 

under International Energy Agency (IEA) Wind Task 30, is 

used for these simulations. This paper reports on this 

comparison study.

POTENTIAL-FLOW THEORY COMPUTER PROGRAMS

Wave loads and motions of offshore structures in waves 

can be calculated by applying potential-flow theory. In linear 

potential theory, the velocity potential and fluid pressure on the 

submerged surface of a body are solved using the boundary 

element method (BEM). Separate solutions are carried out 

simultaneously for the diffraction problem, giving the effects of 

incident waves on the body, and the radiation problem, giving 

the effects of the motion of the body for each of the prescribed 

modes of motion. Each contribution of the first-order solution 

can be calculated distinctly and independently from the others.

DIFFRAC (developed by MARIN) and WAMIT (developed by 

the Massachusetts Institute of Technology) are two examples of 

computer programs based on the potential-flow theory.

For the second-order solution, the diffraction and the 

radiation problem can no longer be considered separately. The 

second-order velocity potential can be completely calculated by 

WAMIT [2], whereas it is approximated by DIFFRAC using 

first-order quantities as described by Pinkster [3]. WAMIT has 

the capability of representing the geometry of the structure by a 

high-order method, whereby the potential is represented by 

continuous B-splines. In DIFFRAC the panels are flat and the 

value of the velocity potential is assumed constant over each 

panel area. 

SECOND-ORDER WAVE LOADS FROM POTENTIALFLOW THEORY

According to Pinkster [3], the second-order wave forces 

can be written as the summation of five different components 

when they are determined by direct pressure integration:

F(2) ������⃗ =

− 1

2 ρρ � ζ (1),rrr

2 . n����

0⃗. dd ���⃗

WW

I

+ 1

2 ρ� ∇φ(1) ∙ ∇φ(1). n����

0⃗ S

dd

II

(1)

+ � ρ.X(1)∙ ∇

∂φ(1)

∂∂ . n����

0⃗dd

S

III

+ Ω��⃗

(1)× M. X⃗̈(1)

IV

+ -ρ ∫∫ ∂φ(2)

∂∂ . n����

0⃗dd V

Subscript (1) denotes when a quantity is of the first order 

and (2) denotes when a quantity is of the second order.

Because components I to IV are quadratic contributions of 

the first-order solution, they can be fully determined from the 

first-order solution. The fifth component involves the secondorder velocity potential that can be calculated by a second-order 

diffraction code applying the perturbation method. As explained 

by López-Pavón, component V is difficult to estimate [4]. It is 

noted that DIFFRAC and WAMIT do not determine this 

component in the same way.

In DIFFRAC, the component V is approximated. Only the 

contribution of the undisturbed incoming wave to the waveexciting force is kept at the second order [3]. In WAMIT, the 

2

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second-order velocity potential includes the contribution of the 

undisturbed incoming wave, as well as the contributions of the 

diffracted and radiated waves.

SIMULATION TOOLS FOR THE DYNAMIC RESPONSE

OF FLOATING FOUNDATIONS FOR WIND TURBINES 

To simulate a floating structure in time, frequencydependent added mass and damping coefficients are 

transformed into inertia coefficients and retardation functions. 

The first-order wave loads are determined by inverse Fourier 

transform of the product of the square root of the wave 

spectrum and the wave-excitation transfer functions. At each 

time step, the equation of motion is solved, taking into account 

the response of the floater and interaction effects between the 

floater and the turbine. Other publications describe how the 

equation of motion of a floating object is implemented in a 

simulation program to make the analysis in the time domain 

possible (see, for example, [5]). aNySIM is such a tool and is 

used to simulate offshore structures. FAST is used to simulate

wind turbines and has been extended to the simulations of 

floating foundations using the same approach as aNySIM [6]. 

FAST accepts the files that are generated by WAMIT as 

hydrodynamic input data. aNySIM uses the hydrodynamic 

database of DIFFRAC as input. In addition to first-order waveexcitation loads, second-order excitation loads can also be 

applied to the floater in aNySIM and FAST through double 

inverse Fourier transforms of the product of wave-amplitude 

pairs and the quadratic transfer function (QTF).

OC4 SEMISUBMERSIBLE

The semisubmersible studied in the OC4 benchmark study 

[7], is used to examine the effects of second-order waveexcitation loads. Some codes in OC4 included second-order 

terms, but these were not the focus of the OC4 project. All 

calculations are repeated twice: once by DIFFRAC and 

aNySIM, and again by WAMIT and FAST. By doing so, this 

study aims at comparing these codes and at better 

understanding the consequences of small differences in the 

calculation process on second-order hydrodynamic effects. 

Although the input description of the OC4 semisubmersible [7] 

is used as reference, some changes are made for the current

study. First, the braces and pontoons are omitted in the 

geometry that is provided to DIFFRAC and WAMIT. Elements 

with small sections compared to the considered wave lengths, 

such as the braces and the pontoons, do not contribute much to 

the diffraction or radiation loads [7]. The OC4 semisubmersible

is sketched in Fig. 1 with braces and pontoons. Figure 2 shows 

the input geometry of DIFFRAC. Second, the viscous damping 

of the OC4 semisubmersible is exclusively brought by 

quadratic damping coefficients for the codes that are based on 

potential-flow theory. In this work some additional damping in 

surge is added to counteract the drift motion more realistically

than only with the quadratic damping of [7]. Table 1 gives the 

values of the damping coefficients. 

FIGURE 1: CONVENTIONS FOR THE SHIP, FIXED 

REFERENTIAL

FIGURE 2: PANEL MESH OF DIFFRAC

TABLE 1: DAMPING VALUES FOR SURGE

Load 

cases

Quadratic 

damping of 

[7] 

(kN⋅m–2⋅s

2

) 

Linear 

damping 

equivalent 

to 

quadratic 

damping 

of [7] 

(kN⋅s/m)

Additional 

linear 

damping 

Total 

damping 

(kN⋅s/m) 

LC2.2 395 40 105 145

The natural periods are important in determining the 

behavior of a floater in waves. They reveal for which wave 

periods the first-order motions will be amplified, and also when 

second-order responses may become visible. Second-order 

loads are by nature perturbations that add to the first-order load, 

which means that they are an order of magnitude smaller by 

definition. The effects of second-order loads on the motions of 

a floater are not normally noticed because they are masked by 

the much bigger first-order motions. Nevertheless, secondorder loads can have large effects on the motions when their 

periods match the natural periods of the floater. Moreover, if 

the damping associated with these periods happens to be small, 

resonance can occur that will induce large motions of the 

floater. The design of the OC4 semisubmersible is optimized 

for a small displacement. First-order responses (in heave, roll, 

3

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and pitch) are just below the wave-energy frequencies. Table 2 

gives the values of the natural periods that are relevant for head 

waves and stern waves.

TABLE 2: NATURAL PERIODS OF THE MOORED 

SEMISUBMERSIBLE

Natural surge/sway period (moored + damped) 113.2 s 

Natural roll/pitch period (moored + damped) 25.1 s

Natural heave period (moored + damped) 17.0 s

LOAD CASES

The load case LC2.2 of the OC4 benchmark study [7] is 

chosen for the simulations. In LC2.2 the semisubmersible is 

exposed to a monodirectional JONSWAP wave spectrum with a 

significant wave height of 6 m, a peak period of 10 s, and a 

peak enhancement factor of 2.87. There is no wind and the 

turbine is not operating. To gain more insight into the results of 

both programs, LC2.2 is split into five simulations in which the 

components of the wave loads are first applied distinctly, 

followed by applying all the components together:

• LC2.2-F1: first-order wave loads

• LC2.2-F2D: second-order difference-frequency loads

• LC2.2-F2DQ: second-order difference-frequency 

loads with only the sum of quadratic terms 

(components I + II + III + IV)

• LC2.2-F2S: second-order sum-frequency loads

• LC2.2-ALL: all components of first- and second-order 

loads together.

LC2.2-ALL is the most realistic simulation because it 

includes first- and second-order wave loads with all possible 

contributions. The other load cases, however, enable the effects 

of specific contributions to be isolated. The two computation

approaches, DIFFRAC plus aNySIM and WAMIT plus FAST, 

are checked against each other, step by step, by running all the 

load cases.

DAMPING

In addition to the radiation contribution of the potentialflow theory to the damping on the semisubmersible, viscous 

loads are added to the hydrodynamic loading. The viscous 

effects are introduced in the model in two distinct ways. First, 

the quadratic drag matrix of [7] is applied. Second, some 

additional linear damping is added for surge so that more 

damping is acting against the slow-drift motion. When the 

floater is exclusively subjected to wave-drift loads (LC2.2-

F2D), the motions are slow and the quadratic drag results in 

very little damping. At the same time, the potential damping in 

surge is also very small in the frequency range of the surge 

eigen mode of the moored semisubmersible. For this purpose, 

some linear damping is added so that the total damping in surge 

matches model test data with the same floating foundation 

reported by Coulling and colleagues [8]. 

The potential damping in surge at the surge natural period 

is very small (0.02 kN⋅s/m) compared to the critical damping 

(2,420 kN⋅s/m). In surge, then, virtually all damping comes 

from viscous drag. So far, the viscous drag has only been 

modeled through a unique quadratic coefficient [7]. The total 

damping (quadratic + linear) should be around 6% of the 

critical damping according to [8] for the range of surge 

amplitudes expected in LC2.2-F2D. As a result, additional 

damping should be added, with its value varying by load case. 

Figure 3 shows the results of decay tests in surge for LC2.2 

with 

• Only the quadratic damping of [7] (plain blue line) 

• The equivalent linear damping (dashed red line)

• The quadratic damping and the additional damping 

(green dashed line)

• Six percent of critical damping as reported in [8] 

(black dot-line).

FIGURE 3: DECAY SIMULATIONS WITH ADDITIONAL 

LINEAR DAMPING TO MATCH TOTAL DAMPING OF 

[8]

A linear damping of 105 kN-s/m is added for LC2.2. The 

same values of quadratic damping and linear damping are used 

in FAST and in aNySIM.

The values specified in [7] are used for the quadratic 

damping in heave (3.88E3 kN⋅m–2⋅s

2

) and pitch (37E6 

kN⋅m⋅s

2

⋅rad–2). No other damping is added in heave and pitch. 

For the sake of the comparison, the QTFs are calculated using 

first-order motions that do not include any damping other than 

the potential damping. As a result, the QTF may be 

overestimated in this study.

CASE STUDY RESULTS 

Waves are following the direction of the surge axis of Fig. 

1. The motions are all determined at the same positions on the 

floater, which are at midship, on the center line, and at s.w.l.

The tensions in the mooring lines are given at the fairleads. In 

4

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this paper, simulation results of FAST using the hydrodynamic 

database calculated by WAMIT are compared to simulation 

results of aNySIM using the hydrodynamic database calculated 

by DIFFRAC.

First, the first-order response of the semisubmersible to the 

waves is checked. In this respect the motion response amplitude 

operators (RAOs) are a powerful way to compare the 

simulations. The comparison is done on a frequency range in 

which the wave energy is large enough. This is achieved with 

good accuracy by calculating the RAOs for frequencies where 

the wave energy spectral density is higher than 5% of its peak. 

RAOs for surge, heave, and pitch are calculated from the results 

of FAST and aNySIM for LC2.2-F1. These RAOs are plotted 

together with those obtained from DIFFRAC (Figs. 4, 5, and 6). 

As expected, the RAOs calculated from the time series of 

aNySIM match the RAOs of DIFFRAC. This confirms that this

simulation is mainly linear. The comparison of FAST and 

aNySIM for the first order is excellent. It is noted that the wave 

energy in LC2.2-F1 is concentrated in a frequency range clearly 

above the surge, heave, and pitch eigen frequencies of the 

semisubmersible.

FIGURE 4: SURGE RAOs FOR LC2.2-F1

FIGURE 5: HEAVE RAOs FOR LC2.2-F1

FIGURE 6: PITCH RAOs FOR LC2.2-F1

Figures 7 and 8 show the results of the simulation for 

LC2.2-F2D where only the low-frequency drift loads are active. 

The mean surge offset is the same in aNySIM (i.e., DIFFRAC) 

and FAST (i.e., WAMIT). This confirms the good agreement of 

WAMIT and DIFFRAC over the main diagonal of the 

difference-frequency QTF. All the motions in aNySIM 

resemble those of FAST with some small differences in 

amplitude and phase (Fig. 7). Although the excitations are 

exclusively of the second order, the amplitudes of the motions 

are clearly visible and even large for surge. The significant 

amplitudes of these motions can be explained by the fact that 

the second-order excitations cover the eigen frequencies in 

surge, heave, and pitch of the moored semisubmersible. The 

response is higher in surge because this motion is weakly 

5

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damped at low frequencies where resonance occurs. Large 

surge oscillations triggered by low-frequency second-order 

wave loads have been reported for tankers [9] and reproduced 

numerically using second-order difference frequency QTFs [3]. 

For this reason, observing these large surge motions here is 

expected. The tensions follow the large surge oscillations, as 

Fig. 8 shows. More remarkably, the heave and pitch motions of 

LC2.2-F2D are also quite large, behavior that is more 

surprising. Nonetheless, Voogt and Soles [10] observed these 

effects in model tests of a semisubmersible with low initial 

stability (i.e., small righting moments). These investigators 

explained that such semisubmersibles are more sensitive to the 

energy in wave groups at frequencies close to the eigen 

frequencies in pitch and roll [10]. With its design optimized for 

small first-order response in heave and pitch, the OC4 

semisubmersible also becomes sensitive to the energy of the 

wave groups. By accounting for the full QTFs in these 

simulations, the long natural periods in heave and pitch of the 

semisubmersible become excited. The difference-frequency 

QTFs result in slow-drift loads with periods around 25 s. They 

can result in even shorter periods that may interfere with the 

first-order response of the semisubmersible.

FIGURE 7: MOTIONS FOR LC2.2-F2D

FIGURE 8: TENSIONS AT FAIRLEADS IN LC2.2-F2D

Figure 9 shows the second-order excitation forces in surge 

in LC2.2-F2DQ on top and in LC2.2-F2D at the bottom. The 

only difference between these two load cases is in the 

component V of Equation (1). In the top graphic, the 

contribution of the second-order velocity potential is 

disregarded (no component V); it is included in the bottom 

graphic. The variation of the amplitude of the wave loads is 

larger when component V is included. Figure 9 also illustrates

that a small difference between aNySIM and FAST arises when 

the contribution of the second-order velocity potential is taken 

into account. This is a consequence of the approximation of the 

component V in DIFFRAC and aNySIM, which differs slightly

from the full potential solution of WAMIT that FAST uses. 

Actually, the effect of the approximation of component V on 

the QTF is clearly visible when the QTF of DIFFRAC is 

plotted together with the QTF of WAMIT (Figs. 15 and 16). 

Parts of the small differences in the motions of aNySIM and 

FAST in LC2.2-F2D find their origin in this approximation of 

component V. 

6

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FIGURE 9: SURGE FORCES FOR LC2.2-F2DQ AND 

LC2.2-F2D

When only the sum-frequency second-order wave loads are 

active, as in LC2.2-F2S, the motions and the variations of the 

tensions are small (Fig. 10). Actually, the sum-frequency 

second-order loads of this spread-moored semisubmersible are 

negligible enough to ignore. Interestingly, the standard 

deviation (SD) of the wave loads is bigger in LC2.2-F2S (SD of 

F2Sx = 620 kN) than in LC2.2-F2D (SD of F2Dx = 265 kN). 

Nevertheless, the responses in motions and tensions are much 

smaller for LC2.2-F2S than for LC2.2-F2D (SD of TenFair2 = 

0.7 kN for LC2.2-F2S against SD of TenFair2 = 62 kN for 

LC2.2-F2D). This demonstrates that second-order loads

become important only when they trigger a resonant behavior

as seen in LC2.2-F2D. Despite some minor phase differences 

between the results of aNySIM and FAST, their results for 

LC2.2-F2S are very similar.

FIGURE 10: RESULTS OF LC2.2-F2S IN SURGE

Figures 11 and 12 show the results of the simulation for 

LC2.2-ALL, where all the wave loads are active (first- and both 

second-order difference- and sum-frequency loads). The 

agreement between FAST and aNySIM is good. Some limited 

differences can be seen in the second-order wave-excitation 

forces and consequently in the motions and the tensions (Figs. 

11 and 12). As observed in LC2.2-F2S and LC2.2-F2D, the 

largest differences in the responses (motions and tensions) are 

caused by the second-order difference-frequency loads, even 

though their amplitude variations are smaller than those of the 

sum-frequency. Figure 12 shows that the two main drivers of 

the tensions in the mooring line 2 are the surge second-order 

difference-frequency force (F2Dx) and the first-order surge 

excitation force (F1x). The large oscillations at slow periods 

corresponding with the natural surge period are directly related 

to F2Dx; the wave-frequency variations are caused by F1x.

This stresses again that second-order excitation should be 

addressed in conjunction with a resonance behavior. 

FIGURE 11: MOTIONS FOR LC2.2-ALL

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FIGURE 12: FORCES FOR LC2.2-ALL

DISCUSSION

The general agreement of the hydrodynamics between

aNySIM and FAST is good. Nevertheless, a detailed 

observation of the results shows that the first-order response in 

LC2.2 agrees better than the second-order response. It is 

interesting to know where the differences arise, even if they are 

small.

A possible cause of discrepancy could reside in the values 

of the QTFs calculated by WAMIT and DIFFRAC. To examine 

this, the main diagonal and the first four side diagonals (based 

on a frequency step of 0.05 rad/s) of the QTFs of WAMIT and 

DIFFRAC are plotted in Figs. 13, 14, 15, and 16. 

The agreement of the QTFs is acceptable despite some 

punctual differences at low frequencies and more systematic 

differences at higher frequencies. As mentioned previously, 

second-order quantities are following from first-order quantities 

in WAMIT and DIFFRAC. For instance, component I of 

Equation (1) is related to the relative wave elevation around the 

floater [3]. This component is largely influenced by the heave 

motion and the pitch (and roll) rotations because they play a 

direct role in the determination of the relative wave height 

around the semisubmersible. The first-order RAOs were

checked in Figs. 4, 5, and 6 for the wave-frequency range of 

LC2.2, not at a higher or lower frequency. Figure 17 shows the 

RAOs over the whole frequency range on which the QTF are 

calculated. Now, these RAOs are determined in the frequency 

domain directly from DIFFRAC and WAMIT. It is clear that 

the amplitudes of the peak in heave and pitch are much bigger 

in WAMIT than in DIFFRAC. This is likely to be caused by 

small differences in the first-order excitation and/or the 

radiation damping at the eigen frequencies. Considering these

discrepancies, the second-order response cannot be expected to 

match at these frequencies. Another discrepancy comes from 

the different way of determining the contribution of the secondorder velocity potential. Although this contribution causes 

differences between DIFFRAC and WAMIT (Figs. 15 and 16), 

they are small.

The analysis of the results shows the importance of 

resonance behavior in conjunction with second-order loading. 

Second-order loads can trigger a resonance amplification if 

their periods match a natural period of the system and if the 

damping is weak at this period for this mode. The OC4

semisubmersible has little damping in surge; therefore, the 

surge drift motion can be important. The frequency range of 

the second-order excitations also needs to cover one or more 

eigen frequencies of the system. The bandwidth of the secondorder load is directly defined by the number of side diagonals 

of the QTF. For the study of drift motions, this bandwidth is 

chosen so that it largely includes the eigen frequencies directly 

related to the mooring system. In the studied load case, the 

focus is on the surge motion with an eigen frequency around 

0.01 Hz. In this way, a bandwidth of 0.04 Hz, corresponding to 

five side diagonals with a step of 0.05 rad/s, seems to be 

enough to trigger the surge response to drift loads. A large part 

of the QTF is neglected, however, when the bandwidth is 

limited to 0.04 Hz (i.e., 0.25 rad/s), as shown in Fig. 18. 

Including fewer diagonals results in decreased wave-excitation 

forces with lower frequency content. To illustrate this 

phenomenon, DIFFRAC and aNySIM were run again with 

difference-frequency QTFs containing fewer side diagonals 

(Fig. 19). It can be observed that including fewer than 20 side 

diagonals (i.e., a bandwidth of less than 1 rad/s) results in the 

exclusion of some of the wave-excitation forces with higher 

frequency content. It also shows that the response to secondorder loads in heave and pitch is clearly visible because enough 

off diagonals were used during the simulations (more than 20 

off diagonals). As explained earlier, the resonance response to a 

second-order excitation is associated to a low damping. In this 

study, only the damping in surge is supported with model-test 

data. In the other modes, the actual values of damping are 

unknown. If the damping in heave and pitch are too low then 

the response in heave and pitch to the second-order differencefrequency loads might be too high; and the other way round. 

Nevertheless, the principle of addressing the responses to 

second-order loads is still valid and this study does it step-bystep. 

8

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
FIGURE 13: AMPLITUDE OF THE MAIN DIAGONAL 

AND THE FIRST FOUR OFF DIAGONALS OF THE 

SURGE DIFFERENCE-FREQUENCY QTF FOR THE 

SUM OF THE QUADRATIC TERMS

FIGURE 14: PHASE OF THE MAIN DIAGONAL AND 

THE FIRST FOUR OFF DIAGONALS OF THE SURGE 

DIFFERENCE-FREQUENCY QTF FOR THE SUM OF 

THE QUADRATIC TERMS

9

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
FIGURE 15: AMPLITUDE OF THE MAIN DIAGONAL 

AND THE FIRST FOUR OFF DIAGONALS OF THE 

SURGE DIFFERENCE-FREQUENCY QTF (kN/m2

) FOR 

THE SECOND-ORDER VELOCITY POTENTIAL TERM

FIGURE 16: PHASE OF THE MAIN DIAGONAL AND 

THE FIRST FOUR OFF DIAGONALS OF THE SURGE 

DIFFERENCE-FREQUENCY QTF (kN/m2) FOR THE 

SECOND-ORDER VELOCITY POTENTIAL TERM

10 This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
FIGURE 17: AMPLITUDE OF HEAVE AND PITCH 

RAOS CALCULATED FROM THE POTENTIAL-FLOW 

BEM RESULTS

FIGURE 18: AMPLITUDE OF DIFFERENCEFREQUENCY QTF FROM DIFFRAC WITH A 

BANDWIDTH OF 2.75 rad/s (INSTEAD OF 0.25 rad/s)

FIGURE 19: EFFECT OF NUMBER OF DIAGONALS IN 

DIFFERENCE-FREQUENCY QTF ON WAVE LOADS

11 This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
SUMMARY AND CONCLUSION

The semisubmersible of the OC4 study was chosen to 

simulate the effect of second-order wave loads with two 

different sets of tools:

• DIFFRAC + aNySIM

• WAMIT + FAST. 

The results of simulations of the same load case (LC2.2) of 

the OC4 study were compared. The effects of the first-order and 

second-order wave loads were isolated for this comparison. 

This comparison showed an overall agreement of all the 

motions. The first-order motions were the same in aNySIM and 

FAST. The surge drift offset was identical and the lowfrequency surge oscillations at the surge natural period of the 

moored semisubmersible were very similar. The second-order 

sum frequency loads appeared to have negligible effects on the 

motions. The effects of the second-order difference-frequency 

loads, however, were large.

The numerical study underlined the importance of 

resonance phenomena in the response to second-order loads. 

For surge, the difference-frequency second-order loads 

dominated the motion response because they excited the 

underdamped surge mode at its very long natural period.

Because the OC4 semisubmersible was designed with a 

relatively small righting moment to obtain a pitch natural 

period just above the most energetic wave periods in 

operational conditions, it became sensitive to second-order 

difference-frequency wave loads in heave and pitch as well as 

in surge. Significant responses in heave and pitch to the 

difference-frequency second-order excitations arose when the 

bandwidths of the QTFs were broad enough. The QTFs were 

calculated for a bandwidth of 3 rad/s in this study, making it 

possible to clearly see the heave and pitch responses to the 

second-order difference-frequency wave loads. 

It is suspected that the lack of damping played a major role 

in the amplification of the motions. The OC4 semisubmersible 

was originally defined with little viscous damping that was 

obtained exclusively through quadratic damping coefficients. 

This additional damping, coming on top of the potential 

damping, was omitted in the determination of the QTFs; it was

only accounted for in the time-domain simulations. The surge 

damping was increased based on published data, but the heave 

and pitch damping were kept as prescribed for the OC4 study.

Adding more damping in surge, heave, and pitch would most 

likely reduce the responses to the second-order excitations that 

have been calculated in this study. Nevertheless, the response to 

vertical second-order wave loads should be considered to avoid 

underestimating the heave and pitch motions in operational 

seas.

ACKNOWLEDGMENTS

The authors are obliged to the participants of the OC4 

project, which operates under IEA Wind Task 30, and to the 

authors of the associated reports. The reports contained

valuable input data for this study. The detailed explanations of 

Tim Bunnik of MARIN on the program DIFFRAC were very 

much appreciated.

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-08-GO28308 with the 

National Renewable Energy Laboratory.

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12 This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.